A note on the essential numerical range of block diagonal operators

Abstract

Abstract In this note, we characterize the essential numerical range of a block diagonal operator T = ⊕ i T i {T=\bigoplus_{i}T_{i}} in terms of the numerical ranges { W ⁢ ( T i ) } i {\{W(T_{i})\}_{i}} of its components. Specifically, the essential numerical range of T is the convex hull of the limit superior of { W ⁢ ( T i ) } i {\{W(T_{i})\}_{i}} . This characterization can be simplified further. In fact, we prove the existence of a decomposition of T for which the convex hull is not required.